Multilinear nilalgebras and the Jacobian theorem

Abstract

If a symmetric multilinear algebra is weakly nil, then it is Engel. This result may be regarded as an infinite-dimensional analogue of the well-known Jacobian theorem, which states that if a polynomial mapping has a polynomial inverse, then its Jacobian matrix is invertible. This refines a theorem of Gerstenhaber and partially answers a question posed by Dotsenko.

4 pp. In addition to the text accepted by IJAC, it is shown in this version that a commutative binary 5-Engel algebra with one generator need not be finite-dimensional